On the convergence acceleration of some continued fractions

A well known method for convergence acceleration of a continued fraction K(an/bn) is based on the use of modified approximants Sn(ωn) in place of the classical ones Sn(0), where ωn are close to tails f (n) of the continued fraction. Recently (Numer. Algorithms 41 (2006), 297–317), the author proposed an iterative method producing tail approximations whose asymptotic expansion’s accuracy is improved in each step. This method can be applied to continued fractions K(an/bn), where, for sufficiently large n, an and bn are polynomials in n (deg an = 2, deg bn ≤ 1). The purpose of this paper is to extend this idea to the class of continued fractions K(an/bn + a ′ n/b ′ n), where an, a ′ n, bn, b ′ n are polynomials in n (deg an = deg a ′ n,deg bn = deg b ′ n). We give examples involving continued fraction expansions of some mathematical constants, as well as elementary and special functions.